ABSTRACT
Consider a mechanical system consisting of two identical masses m that are free to slide over a frictionless horizontal surface. Suppose that the masses are attached to one another, and to two immovable walls, by means of three identical light horizontal springs of spring constant k, as shown in Figure 3.1. The instantaneous state of the system is conveniently specified by the displacements of the left and right masses, x1(t) and x2(t), respectively. The extensions of the left, middle, and right springs are x1, x2 − x1, and −x2, respectively, assuming that x1 = x2 = 0 corresponds to the equilibrium configuration in which the springs are all unextended. The equations of motion of the two masses are thus
m .. x1 = −k x1 + k (x2 − x1), (3.1)
m .. x2 = −k (x2 − x1) + k (−x2). (3.2)
Here, we have made use of the fact that a mass attached to the left end of a spring of extension x and spring constant k experiences a horizontal force +k x, whereas a mass attached to the right end of the same spring experiences an equal and opposite force −k x.
